Noise enhanced detection

Abstract

Performance of some suboptimal detectors can be improved by adding independent noise to their measurements. Improving the performance of a detector by adding a stochastic signal to the measurement can be considered in the framework of stochastic resonance (SR), which can be regarded as the observation of “noise benefits” related to signal transmission in nonlinear systems. Such noise benefits can be in various forms, such as a decrease in probability of error, or an increase in probability of detection under a false-alarm rate constraint. The main focus of this thesis is to investigate noise benefits in the Bayesian, minimax and Neyman-Pearson frameworks, and characterize optimal additional noise components, and quantify their effects. In the first part of the thesis, a Bayesian framework is considered, and the previous results on optimal additional noise components for simple binary hypothesis-testing problems are extended to M-ary composite hypothesis-testing problems. In addition, a practical detection problem is considered in the Bayesian framework. Namely, binary hypothesis-testing via a sign detector is studied for antipodal signals under symmetric Gaussian mixture noise, and the effects of shifting the measurements (observations) used by the sign detector are investigated. First, a sufficient condition is obtained to specify when the sign detectorbased on the modified measurements (called the “modified” sign detector) can have smaller probability of error than the original sign detector. Also, two suf- ficient conditions under which the original sign detector cannot be improved by measurement modification are derived in terms of desired signal and Gaussian mixture noise parameters. Then, for equal variances of the Gaussian components in the mixture noise, it is shown that the probability of error for the modified detector is a monotone increasing function of the variance parameter, which is not always true for the original detector. In addition, the maximum improvement, specified as the ratio between the probabilities of error for the original and the modified detectors, is specified as 2 for infinitesimally small variances of the Gaussian components in the mixture noise. Finally, numerical examples are presented to support the theoretical results, and some extensions to the case of asymmetric Gaussian mixture noise are explained. In the second part of the thesis, the effects of adding independent noise to measurements are studied for M-ary hypothesis-testing problems according to the minimax criterion. It is shown that the optimal additional noise can be represented by a randomization of at most M signal values. In addition, a convex relaxation approach is proposed to obtain an accurate approximation to the noise probability distribution in polynomial time. Furthermore, sufficient conditions are presented to determine when additional noise can or cannot improve the performance of a given detector. Finally, a numerical example is presented. Finally, the effects of additional independent noise are investigated in the Neyman-Pearson framework, and various sufficient conditions on the improvability and the non-improvability of a suboptimal detector are derived. First, a sufficient condition under which the performance of a suboptimal detector cannot be enhanced by additional independent noise is obtained according to the Neyman-Pearson criterion. Then, sufficient conditions are obtained to specifywhen the detector performance can be improved. In addition to a generic condition, various explicit sufficient conditions are proposed for easy evaluation of improvability. Finally, a numerical example is presented and the practicality of the proposed conditions is discussed.